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We prove that a semigroup generated by finitely many truncated convolution operators on Lp[0, 1] with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

A quadratic semigroup algebra is an algebra over a field given by the generators x 1, . . . , x n and a finite set of quadratic relations each of which either has the shape x j x k = 0 or the shape x j x k = x l x m . We prove that a quadratic semigroup algebra given by n generators and $${d\leq \frac{n^2+n}{4}}$$ relations is always infinite dimensional. This strengthens the Golod–Shafarevich estimate for the above class of algebras. Our main result however is that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and δ n relations, where δ n is the first integer greater than $${\frac{n^2+n}{4}}$$ . That is, the above Golod–Shafarevich-type estimate for semigroup algebras is sharp.

We determine the cyclic behaviour of Volterra composition operators, which are defined as $(V_\phif)(x) =\int_0^{\phi(x)}f(t) dt$, $f ? L^p[0, 1]$, 1\leq p where $?$ is a measurable self-map of [0, 1]. The cyclic behaviour of $V_\phi$ is essentially determined by the behaviour of the inducing symbol $\phi$ at 0 and at 1. As a particular result, we provide new examples of quasinilpotent supercyclic operators, which extend and complement previous ones of Hector Salas.

A unitary operator $V$ and a rank $2$ operator $R$ acting on a Hilbert space $\H$ are constructed such that $V+R$ is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic., published in Mathematische Annalen

We construct a bounded linear operator on a separable, reflexive and strictly convex Banach space with the resolvent norm that is constant in a neighbourhood of zero.

Given a Lebesgue measurable self-map \({\varphi}\) of the interval [0, 1], the Volterra- composition operator is defined as \({(V_\varphi)(x)=\int\limits_0^{\varphi(x)}f(t)\,dt, \quad f \in L^p[0,1],\,\, 1\leq p \leq \infty.}\) We develop the spectral theory of these operators. In particular, for a class of natural symbols \({\varphi}\) , finiteness of the spectrum is characterized and formulae for the trace and the convergence exponent of eigenvalues are provided. The positivity of the spectrum as well as the analyticity of the eigenfunctions are also treated. The theory of entire functions as well as solving some Cauchy Problems will play a fundamental role in this theory. We also supply some examples of symbols \({\varphi}\) to which the theory can be applied and, in particular, eigenvalues and eigenfunctions are computed explicitly.

Asymptotic estimates of the norms of orbits of certain operators that commute with the classical Volterra operator V acting on L p [ 0 , 1 ] , with 1 ⩽ p ⩽ ∞ , are obtained. The results apply not only to the Riemann–Liouville operator V r and to I + V r with r > 0 , but also to operators of the form ϕ ( V ) , where ϕ is a holomorphic function at zero. The method to obtain the estimates is based on the fact that the Riemann–Liouville operator as well as the Volterra operator can be related to the Levin–Pfluger theory of holomorphic functions of completely regular growth. Different methods, such as the Denjoy–Carleman theorem, are needed to analyze the behavior of the orbits of I − c V , where c > 0 . The results are applied to the study of cyclic properties of ϕ ( V ) , where ϕ is a holomorphic function at 0.

For an arbitrary associative unital ring $R$, let $J_1$ and $J_2$ be the following noncommutative birational partly defined involutions on the set $M_3(R)$ of $3\times 3$ matrices over $R$: $J_1(M)=M^{-1}$ (the usual matrix inverse) and $J_2(M)_{jk}=(M_{kj})^{-1}\,$ (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich saying that $(J_2\circ J_1)^3$ is the identity map modulo the ${\rm Diag}_{L} \times \rm{Diag}_R$ action $(D_1,D_2)(M)=D_1^{-1}MD_2$ of pairs of invertible diagonal matrices. That is, we show that for each $M$ in the domain where $(J_2\circ J_1)^3$ is defined, there are invertible diagonal $3\times 3$ matrices $D_1=D_1(M)$ and $D_2=D_2(M)$ such that $(J_2\circ J_1)^3(M)=D_1^{-1}MD_2.$, Comment: 28 pages

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any $x\in B$ either $T^nx=0$ for some positive integer n or the sequence $\{T^nx/\|T^nx\|\}$ weakly converges to zero in B . Antisupercyclicity of T means that the angle criterion of supercyclicity is not satisfied for T in the strongest possible way. Normal antisupercyclic operators and antisupercyclic bilateral weighted shifts are characterized. As for the Volterra operator V , it is proved that if $1\leq p\leq\infty$ and any $f\in L_p[0,1]$ then the limit $\lim_{n\to\infty} (n!\|V^nf\|_p)^{1/n}$ does exist and equals $1-\inf\,\hbox{\rm supp}\,(f)$ . Upon using this asymptotic formula it is proved that the operator V acting on the Banach space $L_p[0,1]$ is antisupercyclic for any $p\in(1,\infty)$ . The same statement for $p=1$ or $p=\infty$ is false. The analogous results are proved for operators $V^zf(x)=(1/{\Gamma(z)})\int_0^x f(t) (x-t)^{z-1}\,dt$ when the real part of $z\in{\mathbb C}$ is positive.

The multiplicative spectrum of a complex Banach space X is the class \(\mathcal{K}\)(X) of all (automatically compact and Hausdorff) topological spaces appearing as spectra of Banach algebras (X, *) for all possible continuous multiplications on X turning X into a commutative associative complex algebra with unity. Properties of multiplicative spectra are studied. In particular, we show that \(\mathcal{K}\)(Xn) consists of countable compact spaces with at most n nonisolated points for any separable, hereditarily indecomposable Banach space X. We prove that \(\mathcal{K}\)(C[0, 1]) coincides with the class of all metrizable compact spaces.

As is known, there are everywhere discontinuous infinitely Frechet differentiable functions on the real locally convex spaces and of finitely supported infinitely differentiable functions and, respectively, of generalized functions. In this paper the relationship between the complex differentiability and continuity of a function on a complex locally convex space is considered. We describe a class of complex locally convex spaces, which includes the complex space , such that every Gateaux complex-differentiable function on a space of this class is continuous. We also describe another class of locally convex spaces, which includes the complex space , such that on every space of this class there is an everywhere discontinuous infinitely Frechet complex-differentiable function whose derivatives are continuous.

A study was performed on resonances and time operator for the cusp map. A theorem on spectra of the operator in different function spaces was presented. It was found that the Forbenius-Perron operator has no analytic eigenfunctions corresponding to the eigenvalues different from 0 and 1 while the natural time operator one has infinite-dimensional space of analytic eigenfunctions corresponding to any of its eigenvalues.

We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators Ap, Aac and Asc such that there exists an orthonormal basis of eigenvectors for the operator Ap, the operator Aac has purely absolutely continuous spectrum and the operator Asc has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component Asc into a direct sum of two self-adjoint operators and . The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

We extend the concept of time operator for general semigroups and construct a non-self-adjoint time operator for the diffusion equation which is intertwined with the unilateral shift. We obtain the spectral resolution, the age eigenstates and a new shift representation of the solution of the diffusion equation. Based on previous work we obtain similarly a self-adjoint time operator for Relativistic Diffusion.

We present new general methods to obtain spectral decompositions of dynamical systems in rigged Hilbert spaces and investigate the existence of resonances and the completeness of the associated eigenfunctions. The results are illustrated explicitly for the simplest chaotic endomorphism, namely the Renyi map. This paper extends and completes our previous work.

An example is constructed of an infinite-dimensional separable pre-Hilbert space non-homeomorphic to any of its closed hyperplanes.

LetE be a nonnormable Frechet space, and letE′ be the space of all continuous linear functionals onE in the strong topology. A continuous mappingf:E′→E′ such that for anyt0∈ℝ,x0∈E′, the Cauchy problemx=f(x), x(t0)=x0 has no solutions is constructed.

The structure of the set of functions continuous on a segment and nowhere differentiable is investigated.

We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group ${T_t}_{t\in\C^n}$ with holomorphic dependence on the parameter $t$. This result covers those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups ${T_t}_{t\geq 0}$., arXiv admin note: text overlap with arXiv:0810.1192

Let $D$ be the differentiation operator $Df=f'$ acting on the Fr\'echet space $\H$ of all entire functions in one variable with the standard (compact-open) topology. It is known since 1950's that the set $H(D)$ of hypercyclic vectors for the operator $D$ is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sep\'ulveda whether the set $H(D)$ contains (up to the zero function) a non-trivial subalgebra of $\H$ or an infinite dimensional closed linear subspace of $\H$. In the present article both questions are answered affirmatively.

It is shown that for a bounded weighted bilateral shift $T$ acting on $\ell_p(\Z)$ for $1\leq p\leq 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.

We prove that a continuous linear operator $T$ on a topological vector space $X$ with weak topology is mixing if and only if the dual operator $T'$ has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space $\omega$ due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator $T$ on $\omega$, $T\oplus T$ is also hypercyclic.

It is proved that for any separable infinite dimensional Banach space $X$, there is a bounded linear operator $T$ on $X$ such that $T$ satisfies the Kitai Criterion. The proof is based on quasisimilarity argument and on showing that $I+T$ satisfies the Kitai Criterion for certain backward weighted shifts $T$.

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CW-complexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations $d \leq \frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the case for any number of quadratic relations. The particular point where the number of relations is equal to $ \frac{n(n-1)}{2}$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We prove that over any infinite field, the Anick conjecture holds for $d \geq \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results., Comment: 17 pages, to appear in the Proceedings of the Royal Society Edinburgh A

Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fr\'echet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator., Comment: The paper is submitted to Journal of LMS

We provide a precise description of the lattice of invariant subspaces of composition operators acting on the classical Hardy space, whose inducing symbol is a parabolic non-automorphism. This is achieved with an explicit isomorphism between the Hardy space and the Sobolev Banach algebra $W^{1,2}[0,\infty)$ that induces a bijection between the lattice of the composition operator and the closed ideals of $W^{1,2}[0,\infty)$. In particular, each invariant subspace of parabolic non-automorphism composition operator always consists of the closed span of a set of eigenfunctions. As a consequence, such composition operators have no non-trivial reducing subspaces. For the sake of completeness, we also include a characterization of the closed ideals of the Banach algebra $W^{1,2}[0,\infty)$. Although such a characterization is known, the proof we provide here is somehow different.

Let be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem , , with respect to functions . It is proved that if , then for an arbitrary set . It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to , does not belong to .

We give an example of a complete locally convex m-topology on the algebra of infinite differentiable functions on $[0,1]$ which is strictly coarser than the natural Frechet-topology but finer than the topology of pointwise convergence. A similar construction works on the algebra of continuous functions on $[0,1].$ Using this examples we can separate different notions of diffotopy and homotopy.